1. Field of the Invention
This invention generally relates to computer controlled machinery, and more particularly, to a method and apparatus for generating low-harmonic trajectories for computer-controlled machinery.
2. Prior Art
For computer-controlled machinery, a significant source of internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the components of the computer controlled machine must use.
Polynomial and spline curves have been widely used for robot trajectory synthesis. The resulting trajectories, when expressed in Fourier series, generally indicate a considerable number of harmonics with high-frequencies in the joint (i.e., actuator) coordinate system. The problem becomes even greater for robot manipulators constructed with revolute joints, due to the nonlinearity of the kinematic mapping between the task and the joint coordinates. The nonlinearity of the dynamics of robot manipulators with revolute joints also increases the harmonic content of the actuating torques or forces. For higher operating speeds, the higher harmonics present in the actuating torques required for accurate tracking of the trajectory may be well above the dynamic response limitations ("bandwidth") of currently available (even high performance and direct drive) actuators.
In the prior art, low-harmonic trajectories are generated using a fundamental sinusoidal time function and (n-1) number of its harmonics, as seen by the following equation (1) where .omega. denotes the fundamental frequency of the trajectory pattern, t.sub.e denotes the time needed to complete the trajectory pattern, and s.sub.i, c.sub.i denote m-dimensional vectors of constant trajectory parameters; ##EQU1##
The problem with this approach is that it is difficult to select the coefficient vectors s.sub.i and c.sub.i to prescribe the shape of the trajectory, and the trajectory is not compatible with synthesis in a computer aided environment.